Here is a game no casino will ever offer. A fair coin is tossed until it first lands heads. The pot starts at £2 and doubles with every tails: heads on the first toss pays £2, heads on the second pays £4, on the third £8, and so on without limit. One question decides everything — what is a fair price for a ticket? The mathematics gives an outrageous answer. Half the time you win £2, which contributes £1 to the average; a quarter of the time you win £4, another £1; an eighth of the time £8, another £1 again. Every possible round adds exactly £1 of expected value, and there are infinitely many possible rounds. The expected payout is therefore infinite, and a coldly rational gambler should hand over any finite sum — fifty pounds, a million, everything they own — for a single ticket. Nobody would. Asked honestly, most people offer less than £20.
The puzzle carries a famous surname, and the family behind it was the most brilliant and quarrelsome dynasty in the history of mathematics. Nicolaus Bernoulli, nephew of the great Jacob and Johann, posed it in 1713 in a letter to the French mathematician Pierre Rémond de Montmort. For a quarter of a century the finest minds in Europe worried at the problem. The resolution that stuck came from Nicolaus's cousin Daniel Bernoulli, who published his analysis in 1738 in the journal of the Imperial Academy of Sciences in St Petersburg, the city where he had held a post — which is why a Swiss puzzle about a coin wager has carried a Russian name ever since.
A pound is not a pound
Daniel's insight looks obvious only in hindsight: the value of money is not fixed, but depends on who receives it and how much they already have. A ducat, he argued, means far more to a pauper than to a millionaire. What matters to a real human being is not the cash itself but the use — the utility — it can buy, and utility grows ever more slowly as wealth piles up. Bernoulli proposed that utility rises with the logarithm of wealth, so that each doubling of a fortune adds roughly the same increment of satisfaction. Run the St Petersburg sums through that filter and the infinite expectation collapses to a modest finite figure: for a player of ordinary means, a fair ticket price of only a few pounds — which is precisely what instinct had been saying all along.
Daniel was not quite first. The Geneva mathematician Gabriel Cramer had reached the same essential idea in 1728, in a letter to Nicolaus, proposing the square root of wealth rather than the logarithm; Daniel read it later and acknowledged him handsomely. Between them the two men smuggled psychology into mathematics: the value of money became a fact about people rather than about coins. Economists later canonised the idea as diminishing marginal utility, and it now underpins everything from progressive taxation to the entire insurance industry — a policyholder happily accepts a small certain loss, the premium, to escape a huge unlikely one, which is simply the St Petersburg wager run in reverse.
The world pushes back
There is also a blunter objection: the infinite prize is a fiction. The doubling only continues while the bank can pay, and no bank can pay forever. If the house is worth £1,000,000, the run must stop after about twenty rounds, and the game's expected value shrinks to roughly £20; even a bank holding every pound on Earth would justify a stake of only around £40. Because each doubling adds just £1 of value, an astronomically rich opponent still makes for a cheap ticket. The naturalist Comte de Buffon met the paradox with brute experiment: he reportedly had a child play the game 2,048 times, and the average payout came to about five coins per game — nowhere near infinity. And in 1934 the economist Karl Menger showed the trouble ran deeper still: inflate the prizes suitably and a "super-St Petersburg" game defeats even Bernoulli's logarithm, so long as utility is allowed to grow without bound.
The paradox's afterlife has been enormous. When John von Neumann and Oskar Morgenstern rebuilt economics on rigorous mathematical foundations in their 1944 book Theory of Games and Economic Behavior, the machinery they formalised — expected utility theory — was Bernoulli's idea in modern dress. And when Daniel Kahneman and Amos Tversky launched behavioural economics with prospect theory in 1979, their opening move was to revisit exactly this territory: how people actually weigh gains, losses and long odds, as opposed to how raw arithmetic insists they should. A three-hundred-year-old parlour puzzle thus sits at the root of both the economic orthodoxy and its great rebellion.
The coin game itself remains unplayable and unpriced, which is somehow fitting. Its lesson was never really about coins: it was the first rigorous demonstration that value does not live in the numbers — it lives in us.
Quiz nuggets
- Nicolaus Bernoulli posed the paradox in a 1713 letter to the French mathematician Pierre Rémond de Montmort.
- Daniel Bernoulli's 1738 paper in the journal of the St Petersburg academy of sciences gave the paradox its name.
- Gabriel Cramer of Geneva anticipated the solution in 1728, proposing square-root rather than logarithmic utility.
- Each round of the game adds exactly £1 of expected value, making the total expected payout infinite.
- Von Neumann and Morgenstern's 1944 Theory of Games and Economic Behavior formalised expected utility theory.